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        The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

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        Congruence of normalized eigenforms at two primes

        Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}...
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        55 views

        Is the number of newforms in a fixed maximal ideal bounded?

        Let $N\in \mathbb{Z}_{\geq 1}$. Enumerate the set of primes $p_i$ which do not divide $Np$ in ascending order. Let $\mathbb{F}$ be a finite field of characteristic $p$, $\bar{\chi}$ denote the mod $p$...
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        152 views

        Is the following variant of Shafarevich's theorem known?

        Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
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        62 views

        Hodge-Tate weights of etale cohomology groups

        Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...
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        117 views

        Is it expected that the mod $p$ representation determines a normalized Hecke newform of fixed weight for p large enough?

        Mazur's conjecture on the image of Galois representations of Elliptic curves states that for $N$ large enough there is a unique elliptic curve $E$ over $\mathbb{Q}$ giving rise to a fixed mod $N$ ...
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        Local factors determine Weil representations - proof of the Artin representation case

        This post can be seen as a continuation of this post I created on MathOverflow. I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
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        236 views

        Local properties of Galois representations attached to torsion classes

        $\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$ Let $F$ be a number field, and let $\Gamma$ be ...
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        1answer
        65 views

        Steinberg components of local deformation rings

        Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. ...
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        1answer
        178 views

        Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?

        Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...
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        1answer
        89 views

        Comparison of two $GL_N(\mathbb{Z}_\ell)$ Galois representations

        I have a question about comparing two $\ell$-adic Galois representations. Suppose we have two irreducibible Galois representaions $$ \rho_1,\rho_2: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_N(\...
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        114 views

        Prime ideals being finitely-generated implies coherence?

        Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$ $(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
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        1answer
        175 views

        Local factors determine Weil representations - proof of the cyclic case

        I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post. I want to ...
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        578 views

        Is every 3-dim self-dual Galois representation a symmetic square of 2-dim representation?

        Basically, my question as in the title. Here the Galois representation I consider is an $\ell$-adic Galois representation (comes from geometry). And by the word "self-dual" I mean that representation ...
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        188 views

        Galois actions on cohomology rings of algebraic varieties

        Let $k$ be an arithmetic field. Let $G_k$ be its absolute Galois group. $G_k$ is often studied via its linear action on cohomology (etale, crystalline, ...) "groups" of algebraic varieties over $k$. ...
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        1answer
        121 views

        Heights of contracted ideals

        Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...

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